Two functions f and g with domains D and E can be combined arithmetically as
f (x) + g(x),
f (x) − g(x),
f (x)g(x),
and
The domains of the first three will be D ∩ E, where D ∩ E denotes the intersection of the sets D and E:
the set of all numbers which belong to both sets,
while that of the last function will be D ∩ E less all values of x for which g(x) = 0.
Students may practice graphical addition with a
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Arithmetic Combinations of Functions
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f (x)
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g(x)
Java applet .
Transformations of Functions
Vertical & Horizontal Shifts
Suppose the graph of a function y = f (x) is given.
If c is a positive number there are a number of simple ways we may modify the function.
be the graph shifted
up a distance c, and y = f (x) − c will be the graph shifted
operating on the function
outside the brackets causes a
What happens if we perform the same operations
It is fairly clear that y = f (x) + c will
down a distance c.
In other words,
vertical shift.
inside the brackets?
In other words, how are the graphs of y = f (x + c) and y = f (x − c) related to the graph y = f (x)?
The easy way to figure this
out is to let x = 0.
Then y = f (0 + c) = f (0), so the value of f (x) at c is shifted to be the value of f at 0.
Thus the graph is
shifted a distance c to the
left .
Similarly, the graph of y = f (x − c) is that of y = f (x) shifted a distance c to the
right.
In other words, operating on the function
Students may practice graph shifting with a
inside the brackets causes a
Java applet .
1
horizontal shift.
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If c > 1, there some more simple ways we may modify the function.
It is fairly clear that y = cf (x) will be the graph
1
or
magnified
vertically by the factor c, and y = f (x) will be the graph
compressed or
shrunk
c
by the factor c.
What happens if we perform the same operations
inside the brackets?
1
In other words, how are the graphs of y = f (cx) and y = f ( x) related to the graph y = f (x)?
c
to let x = 1. Then y = f (c(1)) = f (c), so the value of f (x) at 1 is shifted to be the value at 1.
factor c
horizontally .
1
1
Similarly, the graph of y = f
x is that of y = f (x) scaled by the factor
horizontally .
c
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stretched
vertically
The easy way to figure this out is
Thus the graph is scaled by the
What happens if c < 0?
To make things simple, we assume c = −1.
Then the graph of y = −f (x) is just that of y = f (x)
reflected about the x-axis, a
vertical effect, while the graph of y = f (−x) is that of y = f (x)
reflected about the y-axis,
a
horizontal effect.
Remember:
Operations
while those
outside the brackets have a
inside the brackets have a
vertical effect,
horizontal effect.
2
Whenever we take a function of a function, we say we are composing them.
Example 1: Suppose f (x) = x 2 and g(x) = x + 1.
2
Then f g(x) = g(x) = (x + 1)2 = x 2 + 2x + 1 is denoted by f ◦ g(x), so that we have
2
f ◦ g(x) = f g(x) = g(x) = (x + 1)2 = x 2 + 2x + 1.
Note that
order is important:
g ◦ f (x) = g(f (x)) = (f (x)) + 1 = x 2 + 1 is quite different!
Definition: the
composition of two functions f and g is defined to be the function f ◦ g defined by the rule
f ◦ g(x) = f g(x)
Students may practice graphical composition with a
Java applet .
It is convenient to think of functions in this context as processing machines connected is series.
x
f
f(x)
g
g(f(x))
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−5 −4 −3 −2 −1 0 1 2 3 4 5
f (x)
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4
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2 1 0 1 2 3 4
g(x)
−3 −2 −1
0
1 2 3 4 3 2 1
f (f (x)) U
4
3
2
1 0 1 0 1 2 3 , where U =“Undefined”.
f (g(x))
4
3
2
1
0 1 2 3 2 1 0
g(f (x)) U
1
2
3
4 3 2 3 4 3 2
g(g(x)) −1
0
1
2
3 4 3 2 3 4 3
6
5
4
3
2
1
-5
-4
-3
-2
0
-1 0
-1
1
2
3
4
-2
-3
The graphs of all four composite functions are made up of straight line segments. Why? They are shown in green:
y
-5
-4
-3
-2
y
6
6
5
5
4
4
3
3
2
2
1
1
-2
-3
1
2
3
4
5
x
-5
f ◦f
-4
-3
-2
0
-1 0
-1
-2
-3
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It is always a good idea to make up a table of values for the functions to be graphed:
0
-1 0
-1
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Example The graph of f is shown in blue, and the graph of g is shown in red. Draw the graphs of f ◦ f , f ◦ g, g ◦ f , and g ◦ g.
1
2
3
4
5
f ◦g
x
5
x
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y
-5
-4
-3
-2
y
6
6
5
5
4
4
3
3
2
2
1
1
0
-1 0
-1
-2
-3
1
2
3
4
5
x
-5
-4
-3
-2
0
-1 0
-1
-2
g◦f
-3
1
2
3
4
5
x
g◦g
Problems like these have become very common on final exams lately, because they test for understanding, not computational ability.
Both skills are, of course, very important.
Compositions of three or more functions:
By f ◦ g ◦ h(x) we mean f g (h(x)) .
Example Let f (x) = x 2 , g(x) = 2x + 1, and h(x) =
x+1
.
x−1
2
x+1
x+1
2
Then (g ◦ h)(x) = 2h(x) + 1 = 2
+ 1, and (f ◦ g ◦ h)(x) = ((g ◦ h)(x)) = 2
+1
x−1
x−1
5